for power converters have proved to be an interesting alternative to other, mostly linear, control techniques. The control objective is usually achieved through an energy reshaping process and by in-jecting damping to modify the dissipation structure of the system. However, a key question that arises during the implementation of the controller is how to tune the various control parameters. From a circuit theoretic perspective, a PBC forces the closed-loop dynamics to behave as if there are artificial resistors—the con-trol parameters—connected in series or in parallel to the real circuit elements. In this paper, a solution to the tuning problem is proposed that uses the classical Brayton–Moser equations. The method is based on the study of a certain “mixed-potential function” which results in quantitative restrictions on the control parameters. These restrictions seem to be practically relevant in terms stability, overshoot and nonoscillatory responses. The theory is exemplified using the elementary single-switch buck and boost converters.
Index Terms—Brayton–Moser equations, controller
commis-sioning, passivity-based control, power converters, tuning.
I. INTRODUCTION
I
N RECENT years passivity-based control (PBC) design for switched-mode power converters has become quite an active area in both the field of system and control theory and power electronics. One particular PBC technique is based on the classical Euler–Lagrange (EL) equations. The applica-tion of EL-based PBC design to single-switch dc-to-dc power converters was first proposed by Sira-Ramírez et al. [13] and is generalized to larger networks, like the coupled-in-ductor C´ uk converter, and three-phase rectifiers and inverters in e.g., [3], [7], and [12]. One of the major advantages of using the EL approach is that the physical structure (e.g., energy, dissipation, and interconnection), including the non-linear phenomena and features, is explicitly incorporated in the model, and thus in the corresponding PBC. This in contrast to conventional techniques that are mainly based on linearized dynamics and corresponding proportional-integral–derivative (PID) or lead–lag control. Since many power converters are nonlinear nonminimum phase systems, controllers stemming from linear techniques are sometimes difficult to tune as to ensure robust performance, especially in the presence of large setpoint changes and disturbances that cause circuit operation to deviate from the nominal point of operation. Therefore, Manuscript received September 14, 2001; revised August 13, 2002 and March 23, 2004. Recommended by Associate Editor Hua Wang.The authors are with the Delft Center for Systems and Control, Delft Uni-versity of Technology, Mekelweg 2, 2628 CD, The Netherlands (e-mail: d.jelt-sema@dcsc.tudelft.nl; j.m.a.scherpen@dcsc.tudelft.nl).
Digital Object Identifier 10.1109/TAC.2004.832236
controller design may be beneficial.
The basic idea behind PBC design is to modify the energy of the system and add damping by modification of the dissipation structure. In the context of EL-based PBC designs for power-converters, two fundamental questions arise.
1) Which variables have to be stabilized to a certain value in order to regulate the output(s) of interest toward a desired equilibrium value? In other words, are the zero-dynamics of the output(s) to be controlled stable1with respect to the
avail-able control input(s), and if not, for which state variavail-ables is it stable?
2) Where to inject the damping and how to tune the various parameters associated to the energy modification and to the damping assignment stage?
It is hard to give a general answer to the first question since we are not able to give explicit formulations of the zero-dy-namics for a general converter structure. Application of PBC to, for example, the boost, buck-boost [13] and the C´ uk [12] converter, leads to an indirect regulation scheme of the output voltage through regulation of the input current. Since there is no general answer to the first question yet, we continue with checking the stability of the zero-dynamics on a case by case basis.
A first attempt to develop some guidelines for adjusting the damping parameters is done by studying the disturbance atten-uation properties and look for upper and lower bounds on these parameters using -gain analysis techniques in [11]. Since the -gain analysis can be argued to be intrinsically conservative and, in case of large converter structures, the necessary calcu-lations may become rather complex, we study a more practical approach. To our knowledge, apart from -gain analysis, there are some recent interesting works revealing Hamiltonian-based results related to tuning (see, for instance, [9]). However, Hamil-tonian-based PBCs differ from EL-based PBCs, as considered herein.
In previous works about EL-based PBC, the location where to add the damping is mainly motivated by the form of the open-loop dissipation structure in the sense that damping is added to those states that do not contain any damping terms a priori. For example, for the boost converter this means that only damping is injected on the input current—called series damping—be-cause the output voltage already contains a damping term due to
1We should emphasize that in general PBC is a technique that aims at energy
shaping and not at imposing any specific behaviors to certain signals (i.e., out-puts to be controlled). However, in some applications (including a large class of power-converters), EL-based PBC designs may lead to a partial system inver-sion. In that case, a thorough study of the zero-dynamics becomes a subproduct of the method. For a detailed discussion, see [7].
the load resistance, e.g., [7] and [13]. The latter control scheme leads to a PBC regulated circuit that is highly sensitive to load variations and also needs an expensive current sensor to mea-sure the inductor current. This holds for many other switching networks too. Recently, in [3] we have proposed a preliminary solution to overcome the load sensitivity problem by using the concept of parallel damping injection.
In this paper, we further develop the preliminary results of [3] and [5] by adopting the Brayton–Moser (BM) equations [1] to analyze a closed-loop PBC scheme. The main contributions can be summarized as follows: In Section II, we briefly introduce the BM equations and accommodate them for the inclusion of controlled switches. The first part of Section III presents the re-formulation of the EL-based PBC design procedure in terms of the BM equations, which will be referred to as BM-based PBC. Since PBC design is based on modification of the physical struc-ture of the circuit, it is not surprising that the BM framework allows an interpretation of the closed-loop dynamics in sim-ilar physical terms, i.e., in terms of inductors, capacitors and resistors. From a circuit-theoretic point of view, the PBC pro-duces a control signal which forces the closed-loop dynamics to act as if there are virtual resistors connected in series and/or in parallel to the real circuit elements [3], [5]. The second part of Section III presents quantitative guidelines concerning the adjustments of the control parameters based on modified ver-sions of the stability theorems proposed in [1]. Additionally, the latter theorems justify the possibility to choose either a series or a parallel damping injection scheme. Illustrative examples using the elementary buck and boost converters, which describe in form and function a large family of power converters, are pre-sented in Sections IV and V. Based on these two case studies, some novel aspects concerning the robustness properties of the parallel damping injection schemes stemming from BM-based PBC are suggested.
II. SWITCHEDBM EQUATIONS
A. Non-Switched Electrical Circuits
Consider an electrical circuit with inductors and capaci-tors. Assume that there are no capacitor-only loops and no in-ductor-only cutsets. In the early sixties, BM have shown [1] that the dynamical behavior of a broad class2of nonlinear electrical
circuits is governed by the following differential equations:
(1)
where denote the currents through
the inductors, and denote the
voltages across the capacitors, respectively, and
is called the mixed-potential function, to be specified later.
The matrices and denote the
inductance and the capacitance matrices, respectively. Notice
2This class covers all topologically complete electrical circuits. A circuit is
called “topologically complete” if it can be described by an independent set of inductor currents and capacitor voltages such that Kirchhoff’s laws are satisfied. For a detailed treatment, the reader is referred to [16].
that the first equation of (1) constitutes Kirchhoff’s voltage law, while the second constitutes Kirchhoff’s current law. Further-more, as argued in [6], (1) do not establish a Lagrangian system in the classical sense, but they can be viewed as some
degen-erate Lagrangian form.
The mixed-potential function captures the interconnection structure, dissipation structure and external signals. A simple procedure to obtain can be summarized as follows.
• First, treat all series resistors and voltage sources as a short circuit, and treat all parallel resistors and current sources as an open circuit. Apply either Kirchhoff’s current law or Kirchhoff’s voltage law to the remaining circuit, i.e.,
determine or , respectively,
where denotes a matrix of appropriate dimensions. • The internal power circulating across the dynamic
ele-ments is represented by .
• Determine the dissipative current-potential that captures the influence of the current-controlled resistors
, i.e.,
• Determine the dissipative voltage-potential that captures the influence of the voltage-controlled resistors (conductors) , i.e.,
• The total supplied power by the (current-controlled) voltage sources and the (voltage-controlled)
cur-rent sources are represented by and ,
respectively.
• Finally, the mixed-potential is determined by combining the potentials obtained in the previous steps as
For ease of notation, in the sequel we will use the more compact notation
(2)
where and .
Remark 1: In contrast to the Lagrangian or Hamiltonian
functions, that are usually defined by the circuit’s total (co-)en-ergy, the mixed-potential consists of terms related to the power circulating in the circuit. However, it is easily seen that the
circuit’s total co-energy, denoted by , forms a
fundamental part of the BM equations, e.g., [4], i.e., we may
replace and in (1) by
respectively. This property is of main importance in the fol-lowing section. Other similarities and dualities between the BM equations and port-Hamiltonian systems can be found in [4].
Fig. 1. Examples. (a) Buck type converter. (b) Boost type converter.
Remark 2: The resistive current and voltage potentials,
and , are often referred to as the resistive
content and co-content, respectively. This terminology was
in-troduced by Millar in the early 1950s; see [1] and the references therein. In case of linear resistors, the content is simply half the dissipated power expressed in terms of the inductor currents, while the co-content is half the dissipated power in terms of the capacitor voltages. On the other hand, from an EL perspective
and may be considered as some generalized
Rayleigh dissipation functions.
B. Switched-Mode Electrical Circuits
For circuits that contain one or more switches, we denote the
switch position(s) by , where
, i.e.,ONorOFF, or in other words is in the discrete set . Depending on the application, redefinition of the switching function may also result in, for example,
. The mixed-potential function is modified to
include switching functions by letting . For
circuits containing a single switch is defined as (3)
with is the mixed-potential
func-tion for the switch posifunc-tion , and is the
mixed-potential function for the switch position . The way the switch enters the potential function as defined in (3) differs from the definition of the switched Rayleigh dissipation func-tion, as defined in [3], [7], and [12], in the sense that here we have used the concept of superposition of the power flows, where in the references the switch in the dissipation would enter via the dissipated energy. Although we have modified the mixed-potential function for the inclusion of one controllable switch, the approach is also suitable for circuits with more than one switch. Inspired by [7], the dynamics of a switched circuit are then expressed by means of a quadruple , called the switched BM parameters
where and
. Consequently, for every admissible switch vector we have a different but unique set of parameters . Let us next consider the BM dynamics of the single-switch buck and boost converters. These converters describe in form and function a large family of power converters and, therefore, we will use them to exemplify the theory throughout this paper.
1) Buck Converter: Consider the buck converter depicted
in Fig. 1(a), where we have defined and .
If we assume that all elements are linear and time-invariant,
, or . The
internal potential is readily found as .
Furthermore, the dissipative current and voltage potentials are , where represents the load conductance. The supplied power by
the voltage source equals and .
Notice that for this circuit only depends on the position of the switch. The equations describing the dynamical behavior of the buck type converter are then given by
(4)
2) Boost Converter: The BM parameters of the boost
con-verter, depicted in Fig. 1(b), are exactly the same as in the pre-vious case, except for the fact that the internal power becomes
a function of the switch position , i.e., ,
and the current potentials now equal and
. The resulting equations for the boost converter are then given by
(5)
C. Pulse-Width Modulation
The switched BM equations are closely related to the average pulse-width modulation (PWM) models, under the condition that the PWM frequency is sufficiently high; see [7] for a detailed discussion in the EL context. This means that is replaced by the average state , representing the average inductor currents and capacitor voltages, and the discrete control is replaced by its duty ratio function vector . For circuits containing a single switch, we thus have the following consistency conditions
and
The averaged potentials can be considered as a weighted
ratio, with weighting parameter , between and .
In the sequel, we will use the average models with denoting the vector of average inductor currents and capacitor voltages,
respectively, and as the duty ratio of the switch, operating in the closed interval [0, 1].
III. BM-BASEDPASSIVITY-BASEDCONTROL
We are now ready to apply passivity based control to the av-erage PWM BM models. As explained in Subsection II-C, the state vector is replaced by its averaged value , representing the average inductor currents and capacitor voltages, and the dis-crete control is replaced by its duty ratio function vector , i.e., we consider switched BM models of the form
(6) Furthermore, without loss of generality, we assume in our devel-opments that the external voltage and current sources are known and constant.
A. Passivity-Based Controller Design
The rationale behind the design of a passivity-based con-troller for switched-mode circuits is to modify the closed-loop co-energy and add damping by modification of the dissipative structure; see [7]. This means that we start by modifying the (av-erage) co-energy function to arrive at a desired closed-loop co-energy function and modify the (average) dissipative
potential . To do this, let define the average
state errors, where represents the desired trajectories for the average inductor currents and capacitor voltages, respectively. Furthermore, if we take as desired closed-loop co-energy
func-tion , the design procedure of the PBC
reduces to first, making a copy of system (6) in terms of , and second, by adding damping in the errors to ensure asymptotic stability, i.e.,
(7)
where is the injected dissipation and
. The resulting closed-loop dissipative potential evaluated in the error states is then defined by
(8) Finally, an explicit definition of the control action is obtained after solving (7) for . Due to the underlying partial system in-version, needs to be solved with respect to a minimum phase state (or states) as discussed in the introduction. This is tanta-mount to setting the minimum phase states to be controlled to their desired values, and solve for the control with respect to the remaining states. Hence, if we use in (6) together
with (7), and and are quadratic in and ,
re-spectively, then the closed-loop error dynamics are in BM form again, i.e.,
(9)
Fig. 2. Closed-loop interpretation of the series damping PBC regulated buck converter. The ideal transformer is included to compensate for the “virtual” voltage drop overR .
Fig. 3. Zero-dynamics for the parallel damping PBC controlled boost converter.
Invoking Lyapunov/LaSalle arguments, asymptotic stability of the closed-loop error dynamics (9) is easily checked by noting that
(10) where represents the total dissipated power of the closed-loop error system, and where the largest invariant set is given by . Notice that in case the closed-loop dissipation
is linear, (see Remark 2).
B. Tuning of the PBC
So far, we have derived the procedure to obtain a PBC strategy in terms of the BM equations, as is developed in [7] based on the EL equations. We note that the design procedure in terms of the BM equations yields exactly the same controllers as one would obtain using an EL description. However, using the BM formulation, the controller is now directly expressed in physi-cally measurable quantities, i.e., currents and/or voltages, while in the latter frameworks the controller is expressed in terms of charges and/or fluxes. Furthermore, it will be shown that the present setting provides us a systemic tool for tuning the PBC controllers.
Interestingly enough, in [1, Ths. 3 and 4, pp. 19 and 21], sta-bility criteria are developed that use the mixed-potential func-tion. These criteria can be used to rule out the existence of self-sustained oscillations. Hence, if we translate the ideas of [1] to our closed-loop setting, where we assume that the closed-loop error system is in BM form (9), we have strong criteria to tune the various control parameters. In other words, we can assign values to the injected dissipation functions to assure a desired
Fig. 4. General feedback representation of a parallel damping PBC regulated power converter.
dynamic behavior in terms of, for example, overshoot and ro-bustness against load variations.
For the closed-loop error mixed-potential function, a quali-tative Lyapunov-based stability condition for the system (9) is
stated as follows. Let , where
and , denote the error-currents
through the inductors and error-voltages across the capacitors. Furthermore, let
denote the (modified) closed-loop resistance and conductance matrices, respectively.
Theorem 1: If is a positive–definite constant matrix, and (11)
with , then for all the solutions of (9) tend to
zero as .3
Remark 3: Of course, in the previous subsection we already
concluded that if the closed-loop error dynamics
converge to zero according to (10). However, Theorem 1 (and also Theorem 2, as stated later) forms a somewhat more con-servative condition to ensure convergence of (9). Moreover, the theorem provides a lower bound on the control parameters to en-sure a “reasonably nice” response in terms of, e.g., overshoot, settling-time, etc. To illustrate this point, consider the lineariza-tion of (9) in the vicinity of the equilibrium point
and , i.e., , where denotes the linearized
system matrix. Based on the linearized system it can be shown, in terms of the complex frequency domain, that if one of the the-orems is satisfied, each eigenvalue of lies either on the real axis (away from the origin) or on a circle in the left-half plane. The radius of this circle can be made arbitrarily small with . The interested reader is referred to [1] for a detailed discussion of this fact.
3Here, the notationkKk denotes the norm of K, defined as kKk =
max f[Kx] Kxg.
Although it is assumed that is constant in the first place, the criterion of Theorem 1 places a constraint on in terms of , and . Therefore, if is not constant, it may be desirable to choose as a function of in order to fulfill (11). Notice that if Theorem 1 is satisfied, stability is guaranteed regardless of ! A similar criterion for the -matrix can be stated as follows.
Theorem 2: If is a positive–definite constant matrix, and (12)
with , then for all the solutions of (9) tend to
zero as .
Detailed proofs for the case when is a constant matrix are
given in [1]. The proofs for and , follow in
a similar way. Stability of for any admissible follows from the fact that satisfies the Lipschitz condition since is bounded, see also [5]. Notice that may be considered as a fine-tuning parameter. The practical relevance of the criteria is illustrated in the following section (see Remark 5), where particular choices of appear to coincide with tuning criteria stemming form linear techniques, as proposed in [3] and [5].
C. Series/Parallel Damping Injection
Apart from the qualitative behavior of the closed-loop system, the criteria of Theorem 1 and Theorem 2 enable us to choose between two different damping injection strategies: Theorem 1 suggests to add damping at all the inductor currents by injecting
series resistances, while the criterion of Theorem 2 suggests to
inject damping at the capacitor voltages by injecting parallel conductances, i.e., according to the theorems it is sufficient to
modify either or by letting
(13) or
(14) respectively. Hence, stability is guaranteed by selecting either
Fig. 5. Typical open-loop start-up response forV = 5V.
Fig. 6. Closed-loop response for different setpointsV based on series damping: capacitor voltage response (top); average injected damping R () (bottom).
(14), to satisfy the criteria of Theorems 1 or 2 for some suitable , respectively. The concept of parallel damping, as it follows from Theorem 2, coincides with the ideas as recently proposed in [3] and [5]. We will come back to this later on.
IV. TUNINGEXAMPLES OF SERIESDAMPEDPBC REGULATEDCONVERTERS
In this section, we consider two illustrative examples of the series damping PBC strategy of Theorem 1. First, we treat an
example where the interconnection matrix is constant using the buck converter. Second, a series damping PBC for the boost converter is developed and its tuning rules are derived. For both converters we assume for simplicity (and without loss of generality) that the source resistance . Again, we point out that the main reason for studying these two converters is that they describe in form and function a large family of power converter structures. A complicating property of the boost converter, as for many more converter structures, is that due to the nonlinear behavior of the conversion ratio the converter’s natural resonance frequency is varying with the
Fig. 7. Closed-loop response for different setpointsV based on parallel damping: capacitor voltage response (top); average injected damping 1=G () (bottom).
desired output voltage4. This means that the tuning criterion
will also depend on the desired output voltage.
A. Buck Converter
Consider the average BM dynamics of the PWM controlled buck converter (4) derived in Section II. In average mode the state is represented by and the control by . The control objec-tive is to regulate the capacitor voltage toward a desired value, say , without any overshoot. Furthermore, we first as-sume that is perfectly known. As demonstrated in [13], both direct and indirect regulation is possible because the zero-dy-namics for both and , with respect to the control , are
stable. Suppose we set (series
damping injection), then the closed-loop error dynamics satisfy
(15) which is accomplished by the control law
. For a detailed derivation, see [13]. In order to ensure a nonoscillatory asymptotically stable re-sponse, Theorem 1 has to be satisfied. The dissipative part of the mixed-potential of the closed-loop error system (15)
is set to and .
4This means that for every admissible = , the converter exhibits
a different (driving point) impedance.
Hence, by noting that , condition (11) leads to
and, hence, we obtain
where is restricted to the interval . This places a lower bound on in terms of the storage elements and only, regardless of the load and the control . At this point, it is interesting to remark that the actual closed-loop system has a nice circuit-theoretic interpretation. To see this, consider the closed-loop system
(16)
with . Equation (16) can be interpreted as shown
in Fig. 2. Notice that indeed acts as a virtual series damping resistance.
B. Boost Converter
Let us next study the series PBC of the boost converter. For that, we aim at a closed-loop dissipation potential yielding the closed-loop error dynamics
As already shown in [7], this is accomplished by the control law , where the desired inductor
current is given by and is the solution of
the nonlinear differential equation
. In a similar fashion as for the buck converter, a lower bound on is found by applying Theorem 1. In this
case, depends on the control , i.e., , and
the conditions for the theorem are straightforwardly checked to hold. Thus, Theorem 1 is satisfied if
Hence, for a given , if is changing from one setpoint to another, we have for every a different value for , i.e.,
.
C. Robustness and Load Perturbations
Unfortunately, the series damping scheme is highly sensitive to unmodeled changes of the load. To see this, consider for ex-ample the series damping PBC regulated buck converter. Sup-pose that the load is unknown, but bounded, i.e.,
. Decompose the load into a nominal value and a bounded
uncertainty , i.e., . The closed-loop
dy-namics (16) then change to
(17)
with , since we use the nominal value in the
control. Now, the equilibria of (17) are obtained as
from which it is seen that if , the output capacitor voltage will not converge to its desired value . Similar ar-guments hold for the series damping controlled boost converter.
Remark 4: In order to deal with the load uncertainties, the
controller can be extended with an adaptive mechanism to com-pensate for them [7]. A major disadvantage of this method is that the resulting controllers become quite involved, even for simple systems like the boost converter. Another problem that arises is how to tune the adaptive controllers as to ensure stability and nonoscillatory responses. A simple solution to prevent the use of an adaptive mechanism is presented in the following section.
V. PARALLELDAMPINGPBC
In this section, we introduce the concept of parallel damping injection. As is done for the series damping injecting controllers, we again use the buck and boost type converters to illustrate the rationale of the approach. It is shown that this concept has some advantages in contrast to the series damping injection strategy, since it provides an easier solution to preserve the desired equi-librium in case of an unknown load. Moreover, it is shown that parallel damping injection also enables us to regulate a nonmin-imum phase circuit by measuring its nonminnonmin-imum phase output only.
A. Buck Converter
The parallel damping scheme is accomplished as follows. In-stead of injecting damping in the undamped state , we aim at voltage controlled resistors only, resulting in a dissipation po-tential of the closed-loop error system
(18) where we have assumed to have only knowledge of the nominal open-loop load. Following again the procedure of Section III, one possible controller that achieves the stabilization task can
be derived as . (Notice that can be
obtained by measuring the capacitor current .) In this
case, the unperturbed (i.e., ) closed-loop
error dynamics satisfy
and, hence, the tuning criterion for the injected damping yields
(19) Another way to obtain a lower bound on is recently pro-posed in [3]. This method uses the impedance properties of the storage elements, and , to find a precise match with the load conductance . For the parallel damping PBC regulated buck converter we need to consider the closed-loop error equa-tion for the average capacitor voltage , i.e.,
where is the resonance frequency and is the damping factor. From classical control theory, we know [8] that in order to have a perfect damping, has to satisfy (critical damping). This is accomplished if
and thus by letting
(20)
where is referred to as the characteristic
impedance of the circuit. If now , then still a nonoscil-latory response is guaranteed as long as the injected damping
satisfies . However, for values , the
response will become sluggish.
Notice that if , the necessary injected damping to satisfy (20) becomes negative, i.e., . Strictly speaking, the controller then provides energy to the circuit and loses its passivity properties. On the other hand, consider the time-derivative of along the trajectories of the
Remark 5: It is easily checked that for , the BM criterion (19) precisely coincides with the characteristic impedance matching criterion (20).
Now, that we have two criteria to tune the control parameter for a known and constant nominal load, let us next study the case that . The actual capacitor voltage dynamics is then obtained by substitution of the control law into (4). After solving for , we obtain
(21) As for the series damping case, it is worth noting that (21) im-plies that there is a (virtual) resistor connected in parallel with the capacitor and the load resistor. For that reason, we may refer to as a virtual parallel damping resistor.
Considering (21), it is directly seen that for ,
independent of . In other words, unlike for the
series damping controller of Section IV, the equilibrium output voltage of the parallel damping controlled buck converter is in-dependent of the load resistor. Summarizing, we have proved the following proposition.
Proposition 1: Consider the system (4) in closed-loop with
the parallel damping PBC . Given
a desired value for the average capacitor voltage , the controller globally exponentially stabilizes the current and voltage trajectories of the average PWM model (4), where
is replaced by , toward and , for every
.
B. Boost Converter
Similar to the buck converter case, the regulation scheme based on parallel damping injection for the boost converter can be summarized as follows. First we make a copy of the circuit dynamics (5), with , in terms of and, in contrast to the series damping philosophy used in [7] and [13], we inject the damping at the voltage coordinate, like in (18). Hence, in order to obtain an internally stable controller we set and solve for the control . This results in the PBC
(22) where , for , is the solution of the nonlinear differ-ential equation
(23) Notice that now the only signal used for feedback is the
non-minimum phase output capacitor voltage (while for a series
(23), locally exponentially stabilizes the current and voltage tra-jectories of the average PWM model (5) toward the equilibrium
and , for every .
Proof: We start by showing that (22)–(23) is a suitable
controller for the stabilization task with respect to the internal stability, i.e., although we only measure the nonminimum phase output variable , the zero-dynamics of the controller remain stable. For that, we proceed by eliminating from (23) by using (22). Then, after some algebraic manipulations we obtain
(24) The zero-dynamics are obtained by letting coincide with its desired value in (24), that is . The phase-plane diagram
of (24), depicted in Fig. 3, shows that for
all is a locally stable equilibrium point, while
is unstable. Instability of corresponds to the fact that if the switch is in theON-position for too long, the current through the inductor increases until the converter blows up. We conclude that the controller, although based on measuring the nonminimum phase output voltage only, is
fea-sible for all in the range .
The proof of local exponential stability for the PBC based parallel damping follows from Theorem 2 and by the fact that is Lipschitz. Furthermore, the proof that is pre-served in the equilibrium in spite of uncertainties in the load is easily seen by considering the equilibria of (5) and (23).
The design of the passivity-preserving control algorithms based on either series or parallel damping injection scheme is carried out for the average PWM models of the buck and boost type converters. If the constant PWM switching frequency is chosen sufficiently high these models will capture the essential dynamic behavior of the converters, and, as a result, the con-trollers are well defined. Although we have only treated two simple examples, the design and tuning methodology is also applicable to a broad class of other power electronic circuits, as long as the average PWM model of such circuit has a BM structure.
VI. SIMULATIONRESULTS
The general closed-loop representation of the parallel damping PBC design philosophy regulating a real power con-verter, i.e., with the actual current and voltage states instead of the average ones, is depicted in Fig. 4. Here denotes the PWM frequency and is a matrix representing the
series and parallel tuning criteria using SIMULINK. We will use a boost converter with the discrete values for the switch. This means that for the series damping injection scheme the only signal used for feedback is the “real” inductor current
, and for the parallel damping scheme we only use the “real” capacitor voltage . The design parameters of the Boost converter are chosen as follows:
, and the PWM switching frequency is set to 50 kHz. The initial conditions are set to
and .
In Fig. 5, a typical open-loop start-up response is shown for the boost converter. The response shows a large overshoot and is highly oscillatory. In Figs. 6 and 7, the responses of the output capacitor voltage are depicted for different setpoints. We ob-serve that for both schemes the controller, with , rapidly stabilizes the capacitor voltages without any overshoot and oscillations. However, the series damping of Fig. 6 does not reach the desired voltage (dashed line), while the par-allel scheme of Fig. 7 reaches the setpoints within 2% accuracy. The steady state error caused by the series damping PBC is due to the fact that the ripple in the inductor current is usually much higher than the ripple in the output voltage. A better accuracy could be obtained by increasing the PWM frequency or by se-lecting a larger inductor. Notice that the ‘undershoot’ in the ca-pacitor voltage is caused by the nonminimum phase nature of the converter.
Furthermore, Fig. 8 shows the closed-loop response for load perturbations. These perturbations are set to , while both schemes are adjusted to a nominal capacitor voltage of 5 V. As expected from the theory, the parallel damping scheme rapidly manages to restore the capacitor voltage to its nominal value, while the series damping scheme does not manage to restore but forces the closed-loop to deviate from the desired voltage.
VII. CONCLUSION
In this paper, the passivity-based controller design procedure for EL systems in the context of power converters is rewritten in terms of the BM equations. Besides the stability theorems pro-posed in this paper, the advantage of this setting is that the states to be used for feedback are directly in terms of physically mea-surable quantities, i.e., currents and voltages. This in contrast to Lagrangian or Hamiltonian systems, where the coordinates are usually the charges and the fluxes, which in most cases can not be measured directly. Additionally, the assignment of parallel damping does in general not involve the use of current sensors
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for their careful reading and their detailed and constructive reviews.
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Dimitri Jeltsema received the B.Sc. degree in
electrical engineering from the Rotterdam School of Engineering, Rotterdam, The Netherlands, and the M.Sc. degree in systems and control engineering from the University of Hertfordshire, Hertfordshire, U.K., in 1996 and 2000, respectively. He is currently working toward the Ph.D. degree at the Delft Center of Systems and Control, Delft University of Tech-nology, The Netherlands.
During his studies, he worked as an Engineer in several electrical engineering companies. During 2002 and 2003, he was a Visiting Student at the Laboratoire de Signaux et Sys-temes (SUPELEC), Paris, France. His research interests are nonlinear circuit theory, power electronics, switched-mode networks and physical modeling, and control techniques. He is a student member of the Dutch Institute of Systems and Control (DISC).
Jacquelien M. A. Scherpen received the M.Sc. and
Ph.D. degrees in applied mathematics from the Uni-versity of Twente, Twente, The Netherlands, in 1990 and 1994, respectively.
Currently, she is an Associate Professor at the Delft Center for Systems and Control of Delft University of Technology, The Netherlands. She has held visiting research positions at the Universite de Compiegne, France, SUPELEC, Gif-sur-Yvette, France, the University of Tokyo, Japan, the Old Do-minion University, Norfolk, VA, and the University of Twente. Her research interests include nonlinear model reduction methods, realization theory, nonlinear control methods, with in particular modeling and control of physical systems with applications to electrical circuits.
Dr. Scherpen is an Associate Editor of the IEEE TRANSACTIONS ON
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